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\section{The non-section curves of degree $\le12$, and the eight points}\label{sec:newcurves}

Corollary~\ref{cor:new} produced $52$ orbits of $\Q$-rational smooth rational curves of degree $7$ to $12$ on $V$ that are not in the literature, as \emph{classes} in $\NS(V_\Q)$. Four of them are sections of a Brudno-type pencil and were parametrised in Section~\ref{sec:sections}; the other $48$ are not sections of any of the three Brudno-type pencils, and nothing in Sections~\ref{sec:sections}--\ref{sec:census} gives their equations. Here we compute, for each of the $48$ classes, the curve itself: a form cutting it out on $V$, generators of its ideal, and a birational parametrisation over $\Q$ by six binary forms. The point of the exercise is Open Problem~(3) of the first version of this paper: whether the eight solutions of Theorem~\ref{thm:census}(iii), which lie on no known curve and on no section, lie on one of these curves. Seven of them do (Theorem~\ref{thm:incidence}).

We name the $48$ curves $\mathrm R7a$, $\mathrm R8a$--$\mathrm R8e$, $\mathrm R9a$--$\mathrm R9d$, $\mathrm R10a$--$\mathrm R10k$, $\mathrm R11a$--$\mathrm R11p$, $\mathrm R12a$--$\mathrm R12k$ in the order of the rows of the class table of Appendix~\ref{app:classes} (R for \emph{residual}: each is the residual intersection of $V$ with a hypersurface containing some of Bremner's lines and conics), skipping the known curves and the four section curves, which keep their names $\mathrm N7$, $\mathrm N10$, $\mathrm N11a$, $\mathrm N11b$ from Table~\ref{tab:sections}. $C_D$ denotes the unique curve in the class $D$ (Lemma~\ref{lem:irr}); its symmetric images $gC_D$ run through the orbit.

\subsection{Equations and parametrisations}

\begin{proposition}\label{prop:residual}
For each of the $48$ classes $D$ there are $k\in\{2,3,4\}$ and a set $\mathcal S_D$ of between $5$ and $16$ distinct lines and conics, stable under complex conjugation, with
\[
kH-D=\sum_{C\in\mathcal S_D}C\qquad\text{in }\NS(V_{\bar\Q}),
\]
listed in Table~\ref{tab:newcurves} \textup($k=2$ for $\mathrm R7a$, $\mathrm R8a$--$\mathrm R8e$, $\mathrm R9d$; $k=4$ for $\mathrm R11c$, $\mathrm R11d$, $\mathrm R11k$, $\mathrm R11n$--$\mathrm R11p$, $\mathrm R12a$, $\mathrm R12b$, $\mathrm R12d$, $\mathrm R12e$, $\mathrm R12g$--$\mathrm R12j$; $k=3$ otherwise\textup). The forms of degree $k$ on $\PP^5$ vanishing on $\bigcup_{C\in\mathcal S_D}C$ are $I(V)_k\oplus\Q f_D$ for a form $f_D$ with rational coefficients, unique up to $I(V)_k$ and scaling, and
\[
\operatorname{div}(f_D|_V)=C_D+\sum_{C\in\mathcal S_D}C .
\]
The ideal of $C_D$ is $\bigl((I(V)+(f_D)):g^\infty\bigr)$ saturated by the irrelevant ideal, where $g$ is the product of a rational linear form vanishing on each rational $C\in\mathcal S_D$ and of $\ell\bar\ell$ for a linear form $\ell$ vanishing on each conjugate pair; it has Hilbert polynomial $dt+1$, and its intersection numbers with the $56$ lines and conics are those of the class $D$. Each $C_D$ is parametrised birationally over $\Q$ by six binary forms of degree $d=\deg D$ without common factor.
\end{proposition}

\begin{proof}[How this was computed]
The expressions $kH-D=\sum C$ were found by integer programming on the $56$ curves ($0/1$ variables, conjugation-stable, minimal number of curves) and re-checked in $\NS$. Since $V$ is a complete intersection, $H^0(\PP^5,\mathcal O(k))\to H^0(V,\mathcal O_V(k))$ is onto, and a form of degree $k$ vanishing on the reduced curves of $\mathcal S_D$ restricts to a section of $\mathcal O_V(kH-\sum C)=\mathcal O_V(D)$, which has $h^0=1$ for the class of an irreducible $(-2)$-curve; so the kernel of the evaluation map has dimension $\dim I(V)_k+1$ ($4$, $19$ or $61$), which was verified in every case, and $f_D$ is any element outside $I(V)_k$ (made integral and primitive after reduction modulo a Gr\"obner basis of $I(V)$). Vanishing on a line is $k+1$ linear conditions over $\Q(i)$, on a conic $2k+1$ (restriction to the plane of the conic modulo its equation); $f_D$ was checked to vanish on no other line or conic. The saturation and the Hilbert polynomials were computed in Singular; the intersection numbers $C_D\cdot C'$ as the degree of the scheme $C_D\cap C'$ over $\Q(i)$, and solving them against the $56$ basis vectors returns exactly $D$. For the parametrisations, two methods were used: for the curves of degree $7$ and $8$, projection to $\PP^2$ from a coordinate $3$-plane, Singular's \texttt{paraPlaneCurve} (a rational parametrisation over $\Q$ existed in all six cases) and lifting by exact interpolation; for degree $\ge9$, where this is too slow, the following. Rational points on $C_D$ were collected from its intersections with the $20$ rational lines and conics and the known parametrised curves; at the smallest one, $p_0$, a formal power series parametrisation $\gamma(\tau)$ of $C_D$ was computed by Newton iteration on four generators of the ideal with independent gradients at $p_0$; with $e$ the smallest integer in $\{2,\dots,5\}$ for which the Hilbert function of the ideal in degree $e$ equals $ed+1$, the forms $g$ of degree $e$ with $g(\gamma(\tau))\equiv0\bmod\tau^{ed-1}$ form, modulo the ideal, a pencil $\langle g_1,g_2\rangle$, and $\varphi=g_1/g_2$ is a function of degree one on $C_D$ (the base divisor $(ed-1)p_0$ cancels); $\varphi$ was normalised by three rational points and the six coordinate forms recovered from $\gamma$ by a Pad\'e-type linear system in degree $\le d$, whose solution space is one-dimensional. Each parametrisation was verified exactly: the three quadrics \eqref{eq:V}, hence \eqref{eq:sextic} and \eqref{eq:k2}, vanish identically; the six forms have degree exactly $d$, no common factor and content $1$; $f_D$ and every generator of the ideal vanish on them; and the intersection numbers with the $56$ lines and conics, computed as degrees of gcd's over $\Q(i)[m,n]$ as in Appendix~\ref{app:comp}, are those of $D$ with $D^2=-2$ (which proves birationality). The coefficients were reduced by M\"obius transformations and $\mathrm{SL}_2(\Z)$-reduction; the largest coefficient is below $10^4$ for $41$ of the $48$ curves and below $10^6$ for all. Appendix~\ref{app:param} prints the parametrisations of degree $\le9$; all $48$ are in the file \texttt{new\_curves.json} accompanying the paper, together with $f_D$ and the ideals.
\end{proof}

For example, $\mathrm R7a$, the unique non-section curve of degree $7$, has
\[
D=(-2,0,0,2,0,0,1,1,1,1,0,0,0),\qquad 2H-D=\Gamma_1+\Gamma_6+\Gamma_7+Q_7+Q_9+Q_{10},
\]
\[
f_D=2xz+2yz-xu-yu+zu-u^2+xv+yv-zv+2uv+v^2+xw+yw-3zw+2uw-w^2 ;
\]
it passes through $\#1$, $\#4$ and $\#5$, and is $\iota_{\Gamma_{10}+\Gamma_{10}}(S(11;7))$, the image of Delorme's curve of degree $11$ under the negation of the Brudno pencil with respect to the section $\Gamma_{10}$ (Appendix~\ref{app:classes}).

\subsection{The eight points}

\begin{theorem}\label{thm:incidence}
Assume \textup{(H)}. None of the $41$ solutions on $V$ lies on a line or a conic of $V$. For each of the $48$ curves $C_D$ and each of the $41$ points, membership in a symmetric image of $C_D$ was decided exactly by evaluating $f_D$ on all $48$ \textup(for $\#2$: $96$\textup) placements of the point, and every positive answer was certified a second time by a parameter value of the parametrisation. For the solutions of Theorem~\ref{thm:census}\textup{(iii)}:
\begin{center}\small\setlength{\tabcolsep}{4pt}\begin{tabular}{@{}r >{\raggedright\arraybackslash}p{7.2cm}@{}}\toprule solution & non-section curves of degree $\le12$ through it\\\midrule
$\#7=\sol{51}{113}{136}{40}{125}{129}$ & $\mathrm R8c$, $\mathrm R8d$, $\mathrm R9b$, $\mathrm R9c$, $\mathrm R10a$, $\mathrm R10c$, $\mathrm R11a$, $\mathrm R11e$, $\mathrm R12b$\\
$\#37=\sol{391}{499}{612}{351}{571}{572}$ & $\mathrm R8a$, $\mathrm R11k$, $\mathrm R12d$\\
$\#41=\sol{365}{579}{710}{314}{653}{669}$ & $\mathrm R8a$, $\mathrm R11b$\\
$\#49=\sol{268}{499}{885}{65}{572}{879}$ & none\\
$\#50=\sol{359}{519}{892}{221}{603}{884}$ & $\mathrm R8e$, $\mathrm R9b$, $\mathrm R10g$, $\mathrm R11i$, $\mathrm R11n$\\
$\#53=\sol{473}{566}{919}{229}{746}{883}$ & $\mathrm R8a$, $\mathrm R11i$\\
$\#58=\sol{258}{751}{991}{94}{813}{971}$ & $\mathrm R8c$, $\mathrm R10c$, $\mathrm R11i$, $\mathrm R11k$\\
$\#71=\sol{573}{815}{1255}{1}{1123}{1143}$ & $\mathrm R11g$\\
\midrule
$\#21=\sol{124}{277}{337}{92}{311}{317}$ & $\mathrm R8e$, $\mathrm R10g$, $\mathrm R10h$, $\mathrm R10i$, $\mathrm R11i$, $\mathrm R11l$, $\mathrm R11o$\\
\bottomrule\end{tabular}\end{center}
In particular seven of the eight solutions on no known curve and on no section lie on a $\Q$-rational smooth rational curve of degree $\le12$ on $V$, and $\#49$ does not. Altogether $40$ of the $41$ solutions on $V$ lie on a $\Q$-rational smooth rational curve of degree $\le12$ \textup($33$ on a known curve or a section curve, and these seven only on non-section curves\textup), and exactly one, $\#49$, lies on none.
\end{theorem}

The incidence was double-checked by a box search $|m|,|n|\le60$ over every parametrisation, matched against the $74$ solutions: it found every exact incidence and nothing else; and the same exact test on the ten known curves and the four section curves reproduces the ``curves'' column of Appendix~\ref{app:table}. The full incidence table of the $48$ curves against the $74$ solutions is in Appendix~\ref{app:classes} (Table~\ref{tab:newcurves}, last column): the curves of degree $8$ are the most populated ($\mathrm R8a$ carries $\#1,31,37,39,41,52,53$), and $\mathrm R12a$ carries no solution with terms $\le1343$.

\begin{remark}
(i) Theorem~\ref{thm:incidence} removes the eight points from the list of ``points of $V$ on no rational curve'', which now has one entry in the census, $\#49$; it lies on fibres of rank $4,4,4$ of the three Brudno-type pencils through it (Theorem~\ref{thm:rankjump}). Whether $\#49$ lies on a $\Q$-rational smooth rational curve of degree $13$ or more, or on a singular rational curve, we do not know; the enumeration of Section~\ref{sec:lattice} stops at degree $12$, and \S\ref{sec:plan} below describes how to search for curves through a given point without enumerating all classes. (ii) The seven points are the first solutions of \eqref{eq:sextic}, \eqref{eq:k2} that are explained only by rational curves on $V$ which are not sections of a Brudno-type pencil (the known parametric families are all sections, Theorem~\ref{thm:known}); $\#7=\sol{51}{113}{136}{40}{125}{129}$, the seventh smallest solution, lies on nine of them. (iii) A solution on $V$ typically lies on several of the $62$ curves of degree $\le12$; the mean over the $41$ points is $4.4$ and the median $3$; $\#1$ lies on $17$ of them and $\#2$ on $23$. On a surface with $\rho(V_\Q)=13$ and infinitely many rational curves this is to be expected for points of small height; the heuristic count of \S\ref{sec:geo-answer} says that a rational curve of degree $d$ carries about $B^{2/d}$ points of height $\le B$.
\end{remark}